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Gregory Giecold is a PhD student at CEA, Saclay. Dmitry.

In this post I will describe recent work on fermionic Schwinger-Keldysh propagators from AdS/CFT. For further details and references see ArXiv: 0904.4869.

A formulation of the AdS/CFT correspondence relates correlators of a quantum field theory at strong coupling to the boundary behaviour of bulk classical supergravity fields in an asymptotically AdS background. For spinor bulk fields and fermionic dual operators, the prescription is embedded in the relation

$\left\langle \exp\left[ \int d^{d}x \left( \bar{\chi}_0 \mathcal{O} + \bar{\mathcal{O}} \chi_0 \right) \right] \right\rangle_{QFT}=\mathcal{Z}_{SUGRA}[\chi_0, \bar{\chi}_0]$ ,

where $\chi_0=\lim_{r \rightarrow \infty} r^{d – \Delta} \Psi$ and $\Delta$ is the scaling dimension of $\mathcal{O}$ , related to the mass Fermionic Schwinger Keldysh propagators from AdS/CFT of the bulk spinor. The boundary lies at $r \rightarrow \infty$ . The early prescription yields Euclidean correlators. In many circumstances standard Feynman diagrams and S-matrices calculations are not adapted. In non-equilibrium settings, interactions cannot be discarded or switched adiabatically or the system might be unstable. All in all, it’s generally not possible to find an asymptotic state and use the LSZ reduction formula. The initial state is known though, so that $\langle in \mid in \rangle$ matrix elements still provide valuable data. On top of that, some systems such as the quark-gluon plasma and condensed matter models are at strong coupling. It would be of interest to find a way of obtaining real-time correlators from AdS/CFT.

But let us first review the Schwinger-Keldysh formalism for real-time propagators in quantum field theory. The Schwinger-Keldysh prescription provides a way to study real-time Green functions by considering a contour in the complexified time plane. Fields “live” on this contour. In some sense, quantum dynamics does the doubling of the degrees of freedom required for describing non-equilibrium states. Alternatively, one could view this forward-backward closed time contour as a way to introduce a fake doubled Fock space. Initial states are defined on the product of the true Fock space and the twin Fock space. A state-vector prescription has been forged for systems which are described by a density matrix (that’s the case of systems at finite temperature, out of equilibrium, etc.). The action underlying a microscopic description of the system, along with the partition functions are now split according to contributions from the four parts of the time contour, with sources $\eta_i$ and fields $\Upsilon_i$ .

Fermionic Schwinger Keldysh propagators from AdS/CFT

Schwinger-Keldysh contour

This way, contour-ordered Green functions are mapped into a matrix

$i G(j,k)=\frac{1}{i^2} \frac{\delta^2 \ln Z\left[ \eta_{1,2}, \bar{\eta}_{1,2} \right] }{\delta \eta_j \delta \eta^{\dagger}_k}=$
$=i\begin{pmatrix}G_{11} & G_{12} \\{}G_{21} & G_{22}\\ \end{pmatrix}$ .

In the operator formalism

$G_{11}(t,\mathbf{x})=– i \langle T \Upsilon(t,\mathbf{x}) \Upsilon^{\dagger}(0) \rangle$ ,
$G_{12}(t,\mathbf{x})=\pm i \langle \Upsilon^{\dagger}(0) \Upsilon (t,\mathbf{x}) \rangle$ ,
$G_{21} (t,\mathbf{x})=– i \langle \Upsilon (t,\mathbf{x}) \Upsilon^{\dagger}(0) \rangle$ ,
$G_{22}(t,\mathbf{x})=– i \langle \hat{T} \Upsilon(t,\mathbf{x}) \Upsilon^{\dagger}(0) \rangle$ .

with the convention that upper signs stand for fermions and lower ones stand for fields obeying the Bose-Einstein statistics.

Fermionic Schwinger Keldysh propagators from AdS/CFT and $\hat{T}$ denote the time-ordering and anti-time-ordering operators.
Those Schwinger-Keldysh correlators are related to the more familiar retarded and advanced Green functions through

$G_R (x – y)=– i \theta(x^0 – y^0) \langle \left\{ \Upsilon(x), \Upsilon^{\dagger}(y) \right] \rangle$ ,
$G_A (x – y)=+ i \theta(y^0 – x^0) \langle \left\{ \Upsilon(x), \Upsilon^{\dagger}(y) \right] \rangle$ .

with the fancy notation $\left\{.,.\right]$ referring to either a commutator or an anticommutator.

Inserting a complete set of states gives the key relations, e.g., for bosons

$G_{11}(k)=\text{Re} G_R(k) + i \coth(\frac{\omega}{2T}) \text{Im} G_R(k)$ ,
$G_{12}(k)=\frac{2i e^{-(\beta – \sigma)\omega}}{1- e^{-\beta \omega}} \text{Im} G_R(k)$ ,
$G_{21}(k)=\frac{2i e^{- \sigma \omega}}{1 – e^{-\beta \omega}} \text{Im} G_R(k)$ ,
$G_{22}(k)=-\text{Re} G_R(k) + i \coth(\frac{\omega}{2T}) \text{Im} G_R(k)$ .

For fermions similar formula hold, with hyperbolic tangents, some sign changes and the Fermi-Dirac distribution starring instead.

When $\sigma=0$ , $G_{21}(k)=G^{>}(k)$ and $G_{12}(k)=G^{<}(k)$ . Since $G_{21}(k) – G_{12}(k) \mid_{\sigma=0}=2i \text{Im} G_R(k)$ , the relation

$G_R(k) – G_A(k)=G^{>}(k) – G^{<}(k)$

holds as required, whatever the quantum statistics of the field.

Now, are those Green functions directly related to some measurable quantities? From its definition, the lower Green function $G^{<}(k)$ is related to the average density of particles in the system or to some current density. One might also define a spectral density from the lower and upper Green functions. When particles in the system are interacting, one can check that $G^{>}(k)$ encodes the average transition probability when an extra particle of momentum Fermionic Schwinger Keldysh propagators from AdS/CFT is added. Besides, switching to retarded-advanced variables, so-called symmetric or Keldysh Green function appear as the sum of $G_{11}$ and $G_{22}$ . They are proportional to the imaginary part of the retarded propagator. Their interpretation is as correlators for stochastic forces experienced by the system.

Real-time correlators in AdS/CFT

Recall that the original prescription in AdS/CFT for finding gauge theory correlators from classical bulk supergravity fields is in Euclidean signature. However it might actually not be possible to perform an analytic continuation to find Minkowski correlators. This would require some knowledge of the Matsubara frequencies. Yet, in many cases the bulk wave equations can only be solved in some restricted frequency limit. Actually, there exists however a prescription, put forward by Son and Starinets in hep-th/0205051, for computing Minkowski signature retarded Green function in AdS/CFT. It involves a choice of in-going boundary conditions. The drawback is that this cannot be obviously generalized to higher point Green functions. So it might still be of interest to try and compute real-time correlators a la Schwinger-Keldysh in an AdS/CFT setting.

A hint on how to achieve this relies on the observation that Penrose diagrams of asymptotically AdS spacetimes with a black hole exhibit two boundaries on which dual gauge theory fields live. On the other hand, fields and Fock spaces are also doubled in the Schwinger-Keldysh formalism.

Fermionic Schwinger Keldysh propagators from AdS/CFT

Kruskal diagram for the AdS-Schwarzschild black hole

This analogy was used by Herzog and Son, cf. hep-th/0212072. They show how the $2\times2$ matrix of two-point correlation functions for a scalar field and its fictitious partner field is reproduced from the AdS dual supergravity action.

The main idea is to study fields on the Kruskal diagram. Kruskal coordinates (usually labelled U and V) are suited to the study of space–times with horizons. The original, asymptotic observer coordinates (such as those which appear in the familiar Schwarzschild metric) behave badly at the horizon. Yet, a space-ship which would happen to cross a black hole horizon would not see anything particular, until, much later, its crew dies in awful circumstances experiencing unbearable tidal forces as they near the true singularity. This physical singularity cannot be removed by any choice of coordinates. But the fake horizon singularity of the early Schwarzschild metric does not appear any more once you switch to Kruskal coordinates. They also provide an extension to extra regions than the initial Schwarzschild metric. Those are the left and lower quadrants in the Penrose diagram pictured below. One can check that the near horizon behaviour of Fourier modes of the solutions to the scalar equation of motion can be expressed in terms of Kruskal coordinates. While initially the field equation was restricted in the right quadrant of the Penrose diagram, it can be extended to the whole diagram. Fields in the L quadrant can be viewed as the doubled fields of those of the R region, with the Schwinger-Keldysh formalism in mind. Besides, when one extends the mode functions to the complex Kruskal coordinates U and V planes, it turns out that positive–frequency solutions to the wave equation are analytic in the lower U and V complex planes. A solution is composed of only negative-frequency modes provided it is analytic in the upper U and V planes.

Only two linear combinations can be built from the modes in each quadrants which meet the above criterium on holomorphicity. These are

$\left\{\begin{array}{ll}u_o=u_{R,o} + \alpha_o u_{L,o} , \\{}u_i=u_{R,i} + \alpha_i u_{L,i}, \end{array} \right.$

where $u_{R,o}$ is an out-going solution to the wave equation which vanishes in the L quadrant. Similarly, $u_{L,i}$ is in-going and vanishes in the R region.

From the analyticity requirement, the in-going and out-going cross-connecting functions $\alpha_i$ and $\alpha_o$ are constrained to be

$\left\{\begin{array}{ll}\alpha_o=e^{\frac{\pi \omega}{2}}, \\{}\alpha_i=e^{-\frac{\pi \omega}{2}}.\end{array} \right.$

The prescription devised by Herzog and Son then consists in expanding a supergravity field in a basis of outgoing Fermionic Schwinger Keldysh propagators from AdS/CFT and in-going modes in the full Kruskal plane. The coefficients multiplying those basis functions are determined from the boundary behaviour of the field in the L and R quadrants.

For the scalar field they consider in their paper, the Bose-Einstein distribution then naturally appears. To compute real-time correlators, one finally just has to insert the mode expansion for the supergravity field back into the boundary supergravity action. The Son and Starinets prescription expressing retarded and advanced Green functions in terms of in-going and out-going solutions is called up in the process, which finally yields the Schwinger-Keldysh propagators. They obey the relations reviewed at the beginning of this post.

Given the recent interest in fermionic operators in AdS/CFT, with possible applications to condensed matter physics, how does the Herzog and Son recipe generalize to fermions? The near-horizon behaviour of solutions $\psi_{i}$ and $\psi_o$ to the Dirac equation in a AdS-Schwarzschild black hole background is such that additional $\sqrt{V}$ or $\sqrt{-U}$ factors appear as compared to the scalar field case. Given the importance of the solutions being analytic in a complexified extension of the Kruskal plane, 0904.4869 shortly reviews the theory of spinors and twistors in curved and complexified space-times mainly developed by Penrose. In particular, it appears that one can choose a basis for expanding dotted spinors $\Psi_-$ , which embodies those extra square root factors with a spinor dyad $\varsigma, \imath$ . By dotted spinor it is meant (in the Fermionic Schwinger Keldysh propagators from AdS/CFT case; there’s a generalization to other dimensions) the negative-chirality Weyl spinor component of a supergravity Dirac spinor $\Psi=\Psi_+ + \Psi_-$ .

Those square root factors provide the key ingredients for generating the Fermi-Dirac distribution. Let us see how this works. As for the scalar field case, the conditions that positive-frequency solutions are analytic in the lower U and V complex planes and negative-energy modes are analytic in their upper counterparts leads to the following linear combinations

$\left\{\begin{array}{ll}\psi_o=\psi_{R,o} + \beta_o \psi_{L,o}, \\{}\psi_i=\psi_{R,i} + \beta_i \psi_{L,i},\end{array} \right.$

with

$\left\{\begin{array}{ll}\beta_o=i e^{\frac{\pi \omega}{2}},\\{}\beta_i=– i e^{- \frac{\pi \omega}{2}}.\end{array} \right.$

They provide a basis for a spinor field defined over the full Kruskal plane of the AdS-Schwarzschild geometry

$\Psi_{-}(r,k)=\sum_k \left[ a(\omega, \mathbf{k}) \psi_o (k, r) + b(\omega, \mathbf{k}) \psi_i (k, r) \right]$ .

A point worth noting is that one does not have to expand the $\Psi_+$ field. Earlier work has established that the leading-order part in an expansion of this field near the boundary must be fixed. One must fix the “position” and leave the “momentum” free to vary in a set of canonically conjugate pairs given by $\bar{\chi}_0$ , the leading order part of $\bar{\Psi}_+$ near the boundary and \psi_0 , the leading-order component of $\Psi_-$ .

The coefficients $a(\omega, \mathbf{k})$ , $b(\omega, \mathbf{k})$ are determined by requiring that $\Psi_-(r,k)$ approaches $\Psi^R_-(k)$ and $\Psi^L_-(k)$ on their respective boundaries. This entails

$a(\omega, \mathbf{k}) \sqrt{-U} \mid_{r_{\partial M}} \varsigma=\frac{1}{e^{\pi \omega}+1} \left[ \Psi^R_- (k)+ e^{\frac{\pi \omega}{2}} \Psi^L_-(k) \right]$ ,
$b(\omega, \mathbf{k}) \sqrt{V} \mid_{r_{\partial M}} \imath=\frac{1}{e^{\pi \omega}+1} \left[ e^{\pi \omega} \Psi^R_-(k) - e^{\frac{\pi\omega}{2}} \Psi^L_-(k) \right]$ .

with the Fermi-Dirac distribution making its way through the algebra.

A key ingredient comes from the effect on spinor fields of time reversal from going to the R-quadrant to the L one. It must also be taken into account when considering the L quadrant part of the boundary action. When the dust settles, taking functional derivatives of the boundary action with respect to boundary R and L spinors yields the Schwinger-Keldysh correlators for a gauge dual fermionic operator. A review on how retarded fermionic propagators are defined in AdS/CFT and how the leading-order spinor components of $\Psi_+$ and $\Psi_-$ are related is best left to the references.

Open questions

In this post, we have focused on the quadratic part of the supergravity action. It would be interesting to compute higher point real-time correlators from higher order components of the action. It might also be of interest to extend to fermions the work of Skenderis and van Rees. As explained in 0902.4010 their work generalizes the Herzog, Son prescription and accounts for some subtle issues. Recently, there has been a sustained interest in fermions from theories with gravity duals. An open problem is to apply the approach exposed in this post to geometries dual to non-relativistic conformal field theories.

6 Comments

Dmitry says:

May 11, 2009 at 9:29 pm

Dear Gregory,

thanks for the nice post, I am sure NEQNET readers will enjoy it as much as I did.

I have a couple of not-too-technical questions:

a) on the contour you use what’s the physical meaning of $i\sigma$ contribution?

b) related to (a) – in formulae for mode expansions you write everywhere dimensionless Matsubara frequencies (like $\exp{}(2\pi\omega)$ ), i.e., you measure frequency in units of the temperature. If I want to recover dimension and temperature dependence, should I write

$\exp{}(2\pi\beta{}\omega{})$

or

$\exp{}(2\pi\sigma{}\omega{})$ ?

c) what if I set from the beginning? In principle, I could discuss non-equilibrium interacting QFT at zero temperature, and the SK contour will consist of two branches – from $t=-\infty$ to $t=+\infty$ and back without any imaginary contribution. Would the mapping to AdS-Schwarzschild with two causally disconnected regions hold in this case?

Cheers,
Dmitry.

Gregory Giecold says:

May 12, 2009 at 9:54 am

Thank you for this guest post, Dmitry.
* About (a), $0<\sigma<\beta$ is introduced to neatly separate fields from their doubler partners. The review paper “Real- and Imaginary-time field theory[...]” by Landsman and van Weert points to the following reference:
“An equivalence class of quantum field theories at finite temperature”, Matsumoto et al., J. Math. Phys. 25, 3076 (1984).
They explain that there is no $\sigma$ -dependence, using energy conservation.
This latter property won’t hold anymore for truly non-equilibrium systems.
Note that $\sigma=\frac{\beta}{2}$ in the gravity calculation matches with a result from the above paper, where this value of $\sigma$ stems from hermiticity of the hamiltonian.
* As for (b), I should habe mentioned that $\pi T=1$ .
* (c): My guess is that one would have to use the Skenderis-van Rees construction with AdS geometry. They show how one can in principle work with any kind of initial state by gluing two Euclidean slices to a Lorentzian slice of a geometry.

- Dmitry says:
  
  May 12, 2009 at 3:06 pm
  
  Dear Gregory,
  
  About (a), $0<\sigma<\beta$ is introduced to neatly separate fields from their doubler partners.
  
  Sorry for being somewhat rude , but the statement does not make any sense to me. If you explicitly write down the Schwinger-Keldysh Lagrangian, a) doublers get automatically separated if the horizontal contour is infinitely long (and the temperature is zero) – so that two halfs of the contour are effectively split out and non-equilibrium theory is reduced to the equilibrium one. b) if the horizontal contour is not infinitely long, doublers are not split out (the cross-correlation function is called Keldysh Green function, it carries information about occupation numbers, non-zero away from vacuum).
  
  Why would you even want to seperate doublers by hands?? I went through Herzog-Son old papers, it seems that in the context of AdS/CFT $\sigma$ appears automatically (and is equal to half the inverse temperature, as you said). It seems that Horowitz and Son don’t quite understand why $\sigma$ has to appear and what is the physical meaning of it, neither do I. But he feeling is that some important physics is hidden behind $\sigma$ (although my intuition of course may be wrong).
  
  Thanks for letting me know about Skenderis-can Rees paper, I did not know about it. By the way, is it apriori clear that if you complexify the boundary of AdS space, AdS/CFT holds for any submanifold of complexified boudary? (It seems to be the basis of their construction.)
  
  Cheers,
  Dmitry.
  
Gregory Giecold says:

May 12, 2009 at 4:49 pm

I certainly don’t want to imply that we “separate the doublers by hand” in the supergravity calculation.
Let’s go back for a moment to the traditional field theory real-time formalism.
It’s true that most treatments of real-time propagators don’t even mention $\sigma$ . That’s because they stick to a particular choice of the complex time contour.
As explained in e.g. “Finite temperature field theory” by A. Das, or the review by Landsman, van Weert, the time path can be generalized with a vertical part of length $\sigma$ .
Matsumoto et al. explain that the quantum field theories specified by different choices of $\sigma$ are equivalent (i.e. no $\sigma$ -dependence) provided that energy conservation holds. That explains why in the Herzog, Son construction what Herzog and Son say about transforming the sources to get any $\sigma$ -dependence of the cross correlators is true, even though the calculation naturally selects the value $\frac{\beta}{2}$ (with hindsight, from energy conservation or hermicity of the Hamiltonian, according to the Matsumoto et al. paper).
The physical meaning of $\sigma=\frac{\beta}{2}$ seems to be energy conservation for the system (which won’t be verified for truly out of equilibrium settings ; but no such situation has been computed in string theory calculations, I fear)
Do you agree? I’ve just quoted results from the Matsumoto et al. paper and the Landsman, van Weert review.

About Skenderis, van Rees, what they do is to discard some region of the Penrose diagrams. Those regions are specified by the values of the or $\tau$ , depending on the path in the contour. No other coordinates are affected.
I’d say it’s fine as the boundary coordinates of gauge theory operators are those of the corresponding supergravity probes, so the correspondence should not suffer.

Regards,

Gregory

- Dmitry says:
  
  May 13, 2009 at 9:48 pm
  
  Dear Gregory,
  
  thanks for the hint! I went trough Matsumoto et al., and indeed, as it seems, the choice $\sigma=\beta/2$ corresponds to the thermal equilibrium situation (!)
  
  I wouldn’t really say that it is related to energy conservation – you might have a conservation of energy but non-equilibrium dynamics – or to hermitian interact. Hamiltonian (as Matsumoto et al. say) – Hamiltonian is always Hermitean unless you turn on the external field.
  
  I guess, what Matsumoto et al. mean by Hermiticity is the following: you expand the field into background part and small fluctuations above it. In the non-equilibrium situation, you treat background contribution as an external field acting on small fluctuations – the corresponding Hamiltonian describing fluctuating part is of course non-Hermitian (since there is no invariance w.r.t. time translations). Note however that before splitting the field into background and fluctuations you have perfectly hermitian Hamiltonian (and overall evolution background+fluctuations is of course unitary).
  
  which won’t be verified for truly out of equilibrium settings ; but no such situation has been computed in string theory calculations, I fear
  
  Yep, and one comes to a surprising conclusion that from AdS-Schwarzschild/CFT one necessarily gets QFT at final temperature and correspondingly at thermal equilibrium. Of course, one can play with linear response theory and FDT, slightly perturbing equilibrium, but it is impossible to describe strong deviations from thermal equilibrium this way.
  
  Thanks again, I’ve learned so much because of your post and this discussion!
  
  By the way, I also invited Balt van Rees to write a post about his paper with Skederis, and he agreed
  
  Cheers,
  Dmitry.

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Fermionic Schwinger-Keldysh propagators from AdS/CFT

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